Respuesta :

SumitRoy07

1. To factorize $x^3-x^2-6x$, we can first factor out $x$ to get $x(x^2-x-6)$. Now, we need to factorize the quadratic expression $x^2-x-6$. This can be done by finding two numbers that multiply to -6 and add to -1. The numbers are -3 and 2, so we can write:

$x^2-x-6 = (x-3)(x+2)$

Therefore, the factorization of $x^3-x^2-6x$ is:

$x^3-x^2-6x = x(x^2-x-6) = x(x-3)(x+2)$

2. To factorize $3x^3-27x^2+24x$, we can first factor out 3x to get $3x(x^2-9x+8)$. Now, we need to factorize the quadratic expression $x^2-9x+8$. This can be done by finding two numbers that multiply to 8 and add to -9. The numbers are -1 and -8, so we can write:

$x^2-9x+8 = (x-1)(x-8)$

Therefore, the factorization of $3x^3-27x^2+24x$ is:

$3x^3-27x^2+24x = 3x(x^2-9x+8) = 3x(x-1)(x-8)$

3. To factorize $8x^3-44x^2+20x$, we can first factor out 4x to get $4x(2x^2-11x+5)$. Now, we need to factorize the quadratic expression $2x^2-11x+5$. This can be done by finding two numbers that multiply to 10 and add to -11. The numbers are -1 and -10, so we can write:

$2x^2-11x+5 = (2x-1)(x-5)$

Therefore, the factorization of $8x^3-44x^2+20x$ is:

$8x^3-44x^2+20x = 4x(2x^2-11x+5) = 4x(2x-1)(x-5)$[tex][/tex]

semsee45

Answer:

[tex]\textsf{1.} \quad {x(x+2)(x-3)[/tex]

[tex]\textsf{2.} \quad 3x(x-1)(x-8)[/tex]

[tex]\textsf{3.} \quad 4x(2x-1)(x-5)[/tex]

Step-by-step explanation:

Question 1

To factorize the given cubic expression x³ - x² - 6x, first factor out the common term x:

[tex]x(x^2-x-6)[/tex]

Now factorize the quadratic expression (x² - x - 6) by using the technique of splitting the middle term.

The product of the coefficient of the leading term and the constant is -6. Therefore, find two numbers that multiply to -6 and sum to -1.

The two numbers are -3 and 2, so rewrite the middle term as -3x + 2x:

[tex]\begin{aligned}x^2-x-6&=x^2-3x+2x-6\\&=x(x-3)+2(x-3)\\&=(x+2)(x-3)\end{aligned}[/tex]

Therefore, the fully factorised expression is:

[tex]\boxed{x(x+2)(x-3)}[/tex]

[tex]\hrulefill[/tex]

Question 2

To factorize the given cubic expression 3x³- 27x² + 24x, first factor out the common term 3x:

[tex]3x(x^2-9x+8)[/tex]

Now factorize the quadratic expression (x² - 9x + 8) by using the technique of splitting the middle term.

The product of the coefficient of the leading term and the constant is 8. Therefore, find two numbers that multiply to 8 and sum to -9.

The two numbers are -8 and -1, so rewrite the middle term as -8x - x:

[tex]\begin{aligned}x^2-9x+8&=x^2-8x-x+8\\&=x(x-8)-1(x-8)\\&=(x-1)(x-8)\end{aligned}[/tex]

Therefore, the fully factorised expression is:

[tex]\boxed{3x(x-1)(x-8)}[/tex]

[tex]\hrulefill[/tex]

Question 3

To factorize the given cubic expression 8x³ - 44x² + 20x​, first factor out the common term 4x:

[tex]4x(2x^2-11x+5)[/tex]

Now factorize the quadratic expression (2x² - 11x + 5) by using the technique of splitting the middle term.

The product of the coefficient of the leading term and the constant is 10. Therefore, find two numbers that multiply to 10 and sum to -11.

The two numbers are -10 and -1, so rewrite the middle term as -10x - x:

[tex]\begin{aligned}2x^2-11x+5&=2x^2-10x-x+5\\&=2x(x-5)-1(x-5)\\&=(2x-1)(x-5)\end{aligned}[/tex]

Therefore, the fully factorised expression is:

[tex]\boxed{4x(2x-1)(x-5)}[/tex]

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